Recently, Tang, Carlet and Tang presented a combinatorial conjecture about binary strings, allowing proving that all balanced functions in some infinite class they introduced have optimal algebraic immunity. Later, Cohen and Flori completely proved that the conjecture is true. These functions have good (provable or at least observable) cryptographic properties but they are not 1-resilient, which represents a drawback for their use as filter functions in stream ciphers. We propose a construction of an infinite class of 1-resilient Boolean functions with optimal algebraic immunity by modifying the functions in this class. The constructed functions have optimal algebraic degree, that is, meet the Siegenthaler bound, and high nonlinearity. We prove a lower bound on their nonlinearity, but as for the Carlet-Feng functions and for the functions mentioned above, this bound is not enough for ensuring a nonlinearity sufficient for allowing resistance to the fast correlation attack. Nevertheless, as for previously found functions with the same features, there is a gap between the bound that we can prove and the actual values computed for small numbers of variables. Our computations show that the functions in this class have very good nonlinearity and also good immunity to fast algebraic attacks. This is the first time that an infinite class of functions gathers all of the main criteria allowing these functions to be used as filters in stream ciphers.
Open Problems Related to Algebraic Attacks on Stream Ciphers
